Mathematical modeling in radiotherapy for cancer: a comprehensive narrative review

Mathematical modeling has long been a cornerstone of radiotherapy for cancer, guiding treatment prescription, planning, and delivery through versatile applications. As we enter the era of medical big data, where the integration of molecular, imaging, and clinical data at both the tumor and patient levels could promise more precise and personalized cancer treatment, the role of mathematical modeling has become even more critical. This comprehensive narrative review aims to summarize the main applications of mathematical modeling in radiotherapy, bridging the gap between classical models and the latest advancements. The review covers a wide range of applications, including radiobiology, clinical workflows, stereotactic radiosurgery/stereotactic body radiotherapy (SRS/SBRT), spatially fractionated radiotherapy (SFRT), FLASH radiotherapy (FLASH-RT), immune-radiotherapy, and the emerging concept of radiotherapy digital twins. Each of these areas is explored in depth, with a particular focus on how newer trends and innovations are shaping the future of radiation cancer treatment. By examining these diverse applications, this review provides a comprehensive overview of the current state of mathematical modeling in radiotherapy. It also highlights the growing importance of these models in the context of personalized medicine and multi-scale, multi-modal data integration, offering insights into how they can be leveraged to enhance treatment precision and patient outcomes. As radiotherapy continues to evolve, the insights gained from this review will help guide future research and clinical practice, ensuring that mathematical modeling continues to propel innovations in radiation cancer treatment.

Continuum topological derivative - a novel application tool for denoising CT and MRI medical images

BACKGROUND

CT and MRI modalities are important diagnostics tools for exploring the anatomical and tissue properties, respectively of the human beings. Several advancements like HRCT, FLAIR and Propeller have advantages in diagnosing the diseases very accurately, but still have enough space for improvements due to the presence of inherent and instrument noises. In the case of CT and MRI, the quantum mottle and the Gaussian and Rayleigh noises, respectively are still present in their advanced modalities of imaging. This paper addresses the denoising problem with continuum topological derivative technique and proved its trustworthiness based on the comparative study with other traditional filtration methods such as spatial, adaptive, frequency and transformation techniques using measures like visual inspection and performance metrics.

METHODS

This research study focuses on identifying a novel method for denoising by testing different filters on HRCT (High-Resolution Computed Tomography) and MR (Magnetic Resonance) images. The images were acquired from the Image Art Radiological Scan Centre using the SOMATOM CT and SIGNA Explorer (operating at 1.5 Tesla) machines. To compare the performance of the proposed CTD (Continuum Topological Derivative) method, various filters were tested on both HRCT and MR images. The filters tested for comparison were Gaussian (2D convolution operator), Wiener (deconvolution operator), Laplacian and Laplacian diagonal (2nd order partial differential operator), Average, Minimum, and Median (ordinary spatial operators), PMAD (Anisotropic diffusion operator), Kuan (statistical operator), Frost (exponential convolution operator), and HAAR Wavelet (time-frequency operator). The purpose of the study was to evaluate the effectiveness of the CTD method in removing noise compared to the other filters. The performance metrics were analyzed to assess the diligence of noise removal achieved by the CTD method. The primary outcome of the study was the removal of quantum mottle noise in HRCT images, while the secondary outcome focused on removing Gaussian (foreground) and Rayleigh (background) noise in MR images. The study aimed to observe the dynamics of noise removal by examining the values of the performance metrics. In summary, this study aimed to assess the denoising ability of various filters in HRCT and MR images, with the CTD method being the proposed approach. The study evaluated the performance of each filter using specific metrics and compared the results to determine the effectiveness of the CTD method in removing noise from the images.

RESULTS

Based on the calculated performance metric values, it has been observed that the CTD method successfully removed quantum mottle noise in HRCT images and Gaussian as well as Rayleigh noise in MRI. This can be evidenced by the PSNR (Peak Signal-to-Noise Ratio) metric, which consistently exhibited values ranging from 50 to 65 for all the tested images. Additionally, the CTD method demonstrated remarkably low residual values, typically on the order of e-09, which is a distinctive characteristic across all the images. Furthermore, the performance metrics of the CTD method consistently outperformed those of the other tested methods. Consequently, the results of this study have significant implications for the quality, structural similarity, and contrast of HRCT and MR images, enabling clinicians to obtain finer details for diagnostic purposes.

CONCLUSION

Continuum topological derivative algorithm is found to be constructive in removing prominent noises in both CT and MRI images and can serve as a potential tool for recognition of anatomical details in case of diseased and normal ones. The results obtained from this research work are highly inspiring and offer great promise in obtaining accurate diagnostic information for critical cases such as Thoracic Cavity Carina, Brain SPI Globe Lens 4th Ventricle, Brain-Middle Cerebral Artery, Brain-Middle Cerebral Artery and neoplastic lesions. These findings lay the foundation for implementing the proposed CTD technique in routine clinical diagnosis.

Noise-resilient deep learning for integrated circuit tomography

X-ray tomography is a non-destructive imaging technique that reveals the interior of an object from its projections at different angles. Under sparse-view and low-photon sampling, regularization priors are required to retrieve a high-fidelity reconstruction. Recently, deep learning has been used in X-ray tomography. The prior learned from training data replaces the general-purpose priors in iterative algorithms, achieving high-quality reconstructions with a neural network. Previous studies typically assume the noise statistics of test data are acquired a priori from training data, leaving the network susceptible to a change in the noise characteristics under practical imaging conditions. In this work, we propose a noise-resilient deep-reconstruction algorithm and apply it to integrated circuit tomography. By training the network with regularized reconstructions from a conventional algorithm, the learned prior shows strong noise resilience without the need for additional training with noisy examples, and allows us to obtain acceptable reconstructions with fewer photons in test data. The advantages of our framework may further enable low-photon tomographic imaging where long acquisition times limit the ability to acquire a large training set.

Acousto-optic holography

Acousto-optic sensing is based on the interaction between sound and light: pressure waves induce density variations, which, in turn, alter the way light propagates in air. Pressure fields are, thus, characterized by measuring changes in light propagation induced by pressure waves. Although acousto-optic sensing provides a way of acquiring acoustic information noninvasively, its widespread application has been hindered by the use of reconstruction methods ill-suited for representing acoustic fields. In this study, an acousto-optic holography method is proposed in which the sound pressure in the near field of a source is captured via acousto-optic sensing. The acousto-optic measurements are expanded into propagating and evanescent waves, as in near-field acoustic holography, making it possible to completely characterize the radiated field noninvasively. An algebraic formulation of the wave expansion enables the use of arbitrary sets of projections. The proposed method is demonstrated experimentally by capturing the acoustic field radiated by a vibrating plate. Accurate holographic reconstructions of the pressure, particle velocity, and intensity fields are obtained using purely optical data. These results are particularly significant for the study of sound fields at mid and high frequencies, where using conventional transducers could perturb the measured field and spatial sampling requirements are challenging.

Fourier Properties of Symmetric-Geometry Computed Tomography and Its Linogram Reconstruction With Neural Network

In this work, we investigate the Fourier properties of a symmetric-geometry computed tomography (SGCT) with linearly distributed source and detector in a stationary configuration. A linkage between the 1D Fourier Transform of a weighted projection from SGCT and the 2D Fourier Transform of a deformed object is established in a simple mathematical form (i.e., the Fourier slice theorem for SGCT). Based on its Fourier slice theorem and its unique data sampling in the Fourier space, a Linogram-based Fourier reconstruction method is derived for SGCT. We demonstrate that the entire Linogram reconstruction process can be embedded as known operators into an end-to-end neural network. As a learning-based approach, the proposed Linogram-Net has capability of improving CT image quality for non-ideal imaging scenarios, a limited-angle SGCT for instance, through combining weights learning in the projection domain and loss minimization in the image domain. Numerical simulations and physical experiments on an SGCT prototype platform showed that our proposed Linogram-based method can achieve accurate reconstruction from a dual-SGCT scan and can greatly reduce computational complexity when compared with the filtered backprojection type reconstruction. The Linogram-Net achieved accurate reconstruction when projection data are complete and significantly suppressed image artifacts from a limited-angle SGCT scan mimicked by using a clinical CT dataset, with the average CT number error in the selected regions of interest reduced from 67.7 Hounsfield Units (HU) to 28.7 HU, and the average normalized mean square error of overall images reduced from 4.21e-3 to 2.65e-3.

An Inverse Approach of Damage Identification Using Lamb Wave Tomography

A pulse laser combined LWT technique with a two-stage reconstruction algorithm was proposed to realize rapid damage location, or even the evaluation of damage size for plate-like structures. Since the amplitude of Lamb waves in propagation is highly sensitive to damage, including inside damage, the change of the attenuation coefficient of Lamb waves in the inspection region was used as a damage index to reconstruct damage images. In stage one, the rough area of the damage was identified by a comparison of the amplitude of the testing signal data and reference data (undamaged state). In stage two, the damage image was reconstructed using an inverse approach based on the least-square method. In order to verify the effectiveness of the proposed rapid approach, experiments on an aluminum plate with a non-penetrating notch and a carbon fiber-reinforced plastic laminated plate with internal delamination induced by a low-velocity impact were carried out. The results show that the notch can be detected with accurate location, and the delamination image can be reconstructed successfully.

Image-Based 2D Re-Projection for Attenuation Substitution in PET Neuroimaging

PURPOSE

In dual modality positron emission tomography (PET)/magnetic resonance imaging (MRI), attenuation correction (AC) methods are continually improving. Although a new AC can sometimes be generated from existing MR data, its application requires a new reconstruction. We evaluate an approximate 2D projection method that allows offline image-based reprocessing.

PROCEDURE

2-Deoxy-2-[18F]fluoro-D-glucose ([18F]FDG) brain scans were acquired (Siemens HR+) for six subjects. Attenuation data were obtained using the scanner's transmission source (SAC). Additional scanning was performed on a Siemens mMR including production of a Dixon-based MR AC (MRAC). The MRAC was imported to the HR+ and the PET data were reconstructed twice: once using native SAC (ground truth); once using the imported MRAC (imperfect AC). The re-projection method was implemented as follows. The MRAC PET was forward projected to approximately reproduce attenuation-corrected sinograms. The SAC and MRAC images were forward projected and converted to attenuation-correction factors (ACFs). The MRAC ACFs were removed from the MRAC PET sinograms by division; the SAC ACFs were applied by multiplication. The regenerated sinograms were reconstructed by filtered back projection to produce images (SUBAC PET) in which SAC has been substituted for MRAC. Ideally SUBAC PET should match SAC PET. Via coregistered T1 images, FreeSurfer (FS; MGH, Boston) was used to define a set of cortical gray matter regions of interest. Regional activity concentrations were extracted for SAC PET, MRAC PET, and SUBAC PET.

RESULTS

SUBAC PET showed substantially smaller root mean square error than MRAC PET with averaged values of 1.5 % versus 8.1 %.

CONCLUSIONS

Re-projection is a viable image-based method for the application of an alternate attenuation correction in neuroimaging.

Direct Fourier Inversion Reconstruction Algorithm for Computed Laminography

Synchrotron radiation computed laminography (CL) was developed to complement the conventional computed tomography as a non-destructive 3D imaging method for the inspection of flat thin objects. Recent progress in hardware at synchrotron sources allows one to record internal evolution of specimens at the micrometer scale and sub-second range but also requires increased reconstruction speed to follow structural changes online. A 3D image of the sample interior is usually reconstructed by the well-established filtered backprojection (FBP) approach. Despite of a great success in the reduction of reconstruction time via parallel computations, the FBP algorithm still remains a time-consuming procedure. A promising way to significantly shorten computation time is to directly perform backprojection in frequency domain (a direct Fourier inversion approach). The corresponding algorithms are rarely considered in the literature because of a poor performance or inferior reconstruction quality resulted from inaccurate interpolation in Fourier domain. In this paper, we derive a Fourier-based reconstruction equation designed for the CL scanning geometry. Furthermore, we outline the translation of the continuous solution to a discrete version, which utilizes 3D sinc interpolation. A projection resampling technique allowing for the reduction of the expensive interpolation to its 1D version is proposed. A series of numerical experiments confirms that the resulting image quality is well comparable with the FBP approach while reconstruction time is drastically reduced.

NUFFT-Based Iterative Image Reconstruction via Alternating Direction Total Variation Minimization for Sparse-View CT

Sparse-view imaging is a promising scanning method which can reduce the radiation dose in X-ray computed tomography (CT). Reconstruction algorithm for sparse-view imaging system is of significant importance. The adoption of the spatial iterative algorithm for CT image reconstruction has a low operation efficiency and high computation requirement. A novel Fourier-based iterative reconstruction technique that utilizes nonuniform fast Fourier transform is presented in this study along with the advanced total variation (TV) regularization for sparse-view CT. Combined with the alternating direction method, the proposed approach shows excellent efficiency and rapid convergence property. Numerical simulations and real data experiments are performed on a parallel beam CT. Experimental results validate that the proposed method has higher computational efficiency and better reconstruction quality than the conventional algorithms, such as simultaneous algebraic reconstruction technique using TV method and the alternating direction total variation minimization approach, with the same time duration. The proposed method appears to have extensive applications in X-ray CT imaging.

Tomographic reconstruction of damage images in hollow cylinders using Lamb waves

Lamb wave tomography (LWT) is a potential and efficient technique for non-destructive tomographic reconstruction of damage images in structural components or materials. A two-stage inverse algorithm proposed by the authors for quickly reconstructing the damage images was applied to hollow cylinders. An aluminum hollow cylinder with an internal surface pit and a Carbon Fiber Reinforced Plastic (CFRP) laminated hollow cylinder with an artificial internal surface damage were used to validate the proposed method. The results show that the present method is capable of successfully reconstructing the images of the above damages in a larger inspection area with much less experimental data compared to some conventional ultrasonic tomography techniques.

Effect of MR contrast agents on quantitative accuracy of PET in combined whole-body PET/MR imaging

PURPOSE

Clinical PET/MR acquisition protocols entail the use of MR contrast agents (MRCA) that could potentially affect PET quantification following MR-based attenuation correction (AC). We assessed the effect of oral and intravenous (IV) MRCA on PET quantification in PET/MR imaging.

METHODS

We employed two MRCA: Lumirem (oral) and Gadovist (IV). First, we determined their reference PET attenuation values using a PET transmission scan (ECAT-EXACT HR+, Siemens) and a CT scan (PET/CT Biograph 16 HI-REZ, Siemens). Second, we evaluated the attenuation of PET signals in the presence of MRCA. Phantoms were filled with clinically relevant concentrations of MRCA in a background of water and (18)F-fluoride, and imaged using a PET/CT scanner (Biograph 16 HI-REZ, Siemens) and a PET/MR scanner (Biograph mMR, Siemens). Third, we investigated the effect of clinically relevant volumes of MRCA on MR-based AC using human pilot data: a patient study employing Gadovist (IV) and a volunteer study employing two different oral MRCA (Lumirem and pineapple juice). MR-based attenuation maps were calculated following Dixon-based fat-water segmentation and an external atlas-based and pattern recognition (AT&PR) algorithm.

RESULTS

IV and oral MRCA in clinically relevant concentrations were found to have PET attenuation values similar to those of water. The phantom experiments showed that under clinical conditions IV and oral MRCA did not yield additional attenuation of PET emission signals. Patient scans showed that PET attenuation maps are not biased after the administration of IV MRCA but may be biased, however, after ingestion of iron oxide-based oral MRCA when segmentation-based AC algorithms are used. Alternative AC algorithms, such as AT&PR, or alternative oral contrast agents, such as pineapple juice, can yield unbiased attenuation maps.

CONCLUSION

In clinical PET/MR scenarios MRCA are not expected to lead to markedly increased attenuation of the PET emission signals. MR-based attenuation maps may be biased by oral iron oxide-based MRCA unless advanced AC algorithms are used.

Reconstructing virus structures from nanometer to near-atomic resolutions with cryo-electron microscopy and tomography

The past few decades have seen tremendous advances in single-particle electron -cryo-microscopy (cryo-EM). The field has matured to the point that near-atomic resolution density maps can be generated for icosahedral viruses without the need for crystallization. In parallel, substantial progress has been made in determining the structures of nonicosahedrally arranged proteins in viruses by employing either single-particle cryo-EM or cryo-electron tomography (cryo-ET). Implicit in this course have been the availability of a new generation of electron cryo-microscopes and the development of the computational tools that are essential for generating these maps and models. This methodology has enabled structural biologists to analyze structures in increasing detail for virus particles that are in different morphogenetic states. Furthermore, electron imaging of frozen, hydrated cells, in the process of being infected by viruses, has also opened up a new avenue for studying virus structures "in situ". Here we present the common techniques used to acquire and process cryo-EM and cryo-ET data and discuss their implications for structural virology both now and in the future.

Hybrid diversification operator-based evolutionary approach towards tomographic image reconstruction

The proposed algorithm introduces a new and efficient hybrid diversification operator (HDO) in the evolution cycle to improve the tomographic image reconstruction and diversity in the population by using simulated annealing (SA), and the modified form of decreasing law of mutation probability. This evolutionary approach has been used for parallel-ray transmission tomography with the head and lung phantoms. The algorithm is designed to address the observation that the convergence of a genetic algorithm slows down as it evolves. The HDO is shown to yield a higher image quality as compared with the filtered back-projection (FBP), the multiscale wavelet transform, the SA, and the hybrid continuous genetic algorithm (HCGA) techniques. Various crossover operators including uniform, block, and image-row crossover operators have also been analyzed, and the latter has been generally found to give better image quality. The HDO is shown to yield improvements of up to 92% and 120% when compared with FBP in terms of PSNR, for 128 × 128 head and lung phantoms, respectively.

Sensitivity of kinetic macro parameters to changes in dopamine synthesis, storage, and metabolism: a simulation study for [¹⁸F]FDOPA PET by a model with detailed dopamine pathway

Quantitative interpretation of brain [¹⁸F]FDOPA PET data has been made possible by several kinetic modeling approaches, which are based on different assumptions about complex [¹⁸F]FDOPA metabolic pathways in brain tissue. Simple kinetic macro parameters are often utilized to quantitatively evaluate metabolic and physiological processes of interest, which may include DDC activity, vesicular storage, and catabolism from (18) F-labeled dopamine to DOPAC and HVA. A macro parameter most sensitive to the changes of these processes would be potentially beneficial to identify impaired processes in a neurodegenerative disorder such as Parkinson's disease. The purpose of this study is a systematic comparison of several [¹⁸F]FDOPA macro parameters in terms of sensitivities to process-specific changes in simulated time-activity curve (TAC) data of [¹⁸F]FDOPA PET. We introduced a multiple-compartment kinetic model to simulate PET TACs with physiological changes in the dopamine pathway. TACs in the alteration of dopamine synthesis, storage, and metabolism were simulated with a plasma input function obtained by a non-human primate [¹⁸F]FDOPA PET study. Kinetic macro parameters were calculated using three conventional linear approaches (Gjedde-Patlak, Logan, and Kumakura methods). For simulated changes in dopamine storage and metabolism, the slow clearance rate (k(loss) ) as calculated by the Kumakura method showed the highest sensitivity to these changes. Although k(loss) performed well at typical ROI noise levels, there was large bias at high noise level. In contrast, for simulated changes in DDC activity it was found that K(i) and V(T), estimated by Gjedde-Patlak and Logan method respectively, have better performance than k(loss).

Systematic approach to cutoff frequency selection in continuous-wave electron paramagnetic resonance imaging

This article describes a systematic method for determining the cutoff frequency of the low-pass window function that is used for deconvolution in two-dimensional continuous-wave electron paramagnetic resonance (EPR) imaging. An evaluation function for the criterion used to select the cutoff frequency is proposed, and is the product of the effective width of the point spread function for a localized point signal and the noise amplitude of a resultant EPR image. The present method was applied to EPR imaging for a phantom, and the result of cutoff frequency selection was compared with that based on a previously reported method for the same projection data set. The evaluation function has a global minimum point that gives the appropriate cutoff frequency. Images with reasonably good resolution and noise suppression can be obtained from projections with an automatically selected cutoff frequency based on the present method.

Simultaneous reconstruction of flow and temperature cross-sections in gases using acoustic tomography

This paper describes the use of air-coupled ultrasonic tomography for the simultaneous measurement of flow and temperature variations in gases. Air-coupled ultrasonic transducers were used to collect through-transmission data from a heated gas jet. A transducer pair was scanned in two-dimensional sections at an angle to the jet, and travel time and amplitude data recorded along various paths in counter-propagating directions. Parallel-beam tomographic reconstruction techniques allowed images to be formed of variations in either temperature or flow velocity. Results have been obtained using heated jets, where it has been shown that it is possible to separate the two variables successfully.

Exact frequency-domain reconstruction for thermoacoustic tomography--II: Cylindrical geometry

Microwave-induced thermoacoustic tomography (TAT) in a cylindrical configuration is developed to image biological tissue. Thermoacoustic signals are acquired by scanning a flat ultrasonic transducer. Using a new expansion of a spherical wave in cylindrical coordinates, we apply the Fourier and Hankel transforms to TAT and obtain an exact frequency-domain reconstruction method. The effect of discrete spatial sampling on image quality is analyzed. An aliasing-proof reconstruction method is proposed. Numerical and experimental results are included.

Exact frequency-domain reconstruction for thermoacoustic tomography--I: Planar geometry

We report an exact and fast Fourier-domain reconstruction algorithm for thermoacoustic tomography in a planar configuration assuming thermal confinement and constant acoustic speed. The effects of the finite size of the detector and the finite length of the excitation pulse are explicitly included in the reconstruction algorithm. The algorithm is numerically and experimentally verified. We also demonstrate that the blurring caused by the finite size of the detector surface is the primary limiting factor on the resolution and that it can be compensated for by deconvolution.

Discretization of the radon transform and of its inverse by spline convolutions

We present an explicit formula for B-spline convolution kernels; these are defined as the convolution of several B-splines of variable widths h(i) and degrees n(i). We apply our results to derive spline-convolution-based algorithms for two closely related problems: the computation of the Radon transform and of its inverse. First, we present an efficient discrete implementation of the Radon transform that is optimal in the least-squares sense. We then consider the reverse problem and introduce a new spline-convolution version of the filtered back-projection algorithm for tomographic reconstruction. In both cases, our explicit kernel formula allows for the use of high-degree splines; these offer better approximation performance than the conventional lower-degree formulations (e.g., piecewise constant or piecewise linear models). We present multiple experiments to validate our approach and to find the parameters that give the best tradeoff between image quality and computational complexity. In particular, we find that it can be computationally more efficient to increase the approximation degree than to increase the sampling rate.

On the optimality of the gridding reconstruction algorithm

Gridding reconstruction is a method to reconstruct data onto a Cartesian grid from a set of nonuniformly sampled measurements. This method is appreciated for being robust and computationally fast. However, it lacks solid analysis and design tools to quantify or minimize the reconstruction error. Least squares reconstruction (LSR), on the other hand, is another method which is optimal in the sense that it minimizes the reconstruction error. This method is computationally intensive and, in many cases, sensitive to measurement noise. Hence, it is rarely used in practice. Despite their seemingly different approaches, the gridding and LSR methods are shown to be closely related. The similarity between these two methods is accentuated when they are properly expressed in a common matrix form. It is shown that the gridding algorithm can be considered an approximation to the least squares method. The optimal gridding parameters are defined as the ones which yield the minimum approximation error. These parameters are calculated by minimizing the norm of an approximation error matrix. This problem is studied and solved in the general form of approximation using linearly structured matrices. This method not only supports more general forms of the gridding algorithm, it can also be used to accelerate the reconstruction techniques from incomplete data. The application of this method to a case of two-dimensional (2-D) spiral magnetic resonance imaging shows a reduction of more than 4 dB in the average reconstruction error.

An optimal and efficient new gridding algorithm using singular value decomposition

The problem of handling data that falls on a nonequally spaced grid occurs in numerous fields of science, ranging from radio-astronomy to medical imaging. In MRI, this condition arises when sampling under time-varying gradients in sequences such as echo-planar imaging (EPI), spiral scans, or radial scans. The technique currently being used to interpolate the nonuniform samples onto a Cartesian grid is called the gridding algorithm. In this paper, a new method for uniform resampling is presented that is both optimal and efficient. It is first shown that the resampling problem can be formulated as a problem of solving a set of linear equations Ax = b, where x and b are vectors of the uniform and nonuniform samples, respectively, and A is a matrix of the sinc interpolation coefficients. In a procedure called Uniform Re-Sampling (URS), this set of equations is given an optimal solution using the pseudoinverse matrix which is computed using singular value decomposition (SVD). In large problems, this solution is neither practical nor computationally efficient. Another method is presented, called the Block Uniform Re-Sampling (BURS) algorithm, which decomposes the problem into solving a small set of linear equations for each uniform grid point. These equations are a subset of the original equations Ax = b and are once again solved using SVD. The final result is both optimal and computationally efficient. The results of the new method are compared with those obtained using the conventional gridding algorithm via simulations.

Quasi-bandlimited properties of radon transforms and their implications for increasing angular sampling densities

The n-dimensional (n-D) radon transform, which forms the mathematical basis for a broad variety of tomographic imaging applications, can be viewed as an n-D function in n-D sinogram space. Accurate reconstruction of continuous or discrete tomographic images requires full knowledge of the radon transform in the corresponding n-D sinogram space. In practice, however, one can have only a finite set of discrete samples of the radon transform in the sinogram space. One often derives the desired full knowledge of the radon transform from its discrete samples by invoking various interpolation algorithms. According to the Wittaker-Shannon sampling theorem, a necessary condition for a full and unique recovery of the radon transform from its discrete samples is that the radon transform itself be bandlimited. Therefore, it is necessary to analyze the bandlimited properties of the radon transform. In this work, we analyze explicitly the bandlimited properties of the radon transform and show that the radon transform is mathematically quasi-bandlimited [or essentially bandlimited] in two quantitative senses and can essentially be treated as bandlimited in practice. The quasi-bandlimited properties can be used for increasing the angular sampling density of the radon transform.

Projection-reconstruction spectroscopic imaging for B0 field plotting and shimming without pulsed gradients

Shimming is important. Noniterative methods are desirable. Such methods exist for shimming a spectrometer with pulsed field gradients, generally based on field maps made by spin-warp Fourier imaging. For spectrometers with no pulsed gradients (or for cases where T2 is too short to permit echo imaging), an alternative method is presented: projection-reconstruction spectroscopic imaging, which can be accomplished using only the shim coils of a conventional spectrometer. Images so acquired can be used to map the field, even in the presence of multiple spectral components. Noniterative optimization of the axial shims of a GN-300 spectrometer is demonstrated using 1D + 1D spectroscopic images. Prospects for extending the technique to include the radial shims using 3D + 1D spectroscopic images are discussed.

Improved automatic off-resonance correction without a field map in spiral imaging

Non-2DFT k-space readout strategies are useful in fast imaging but prone to blurring when reconstructed off resonance. Field inhomogeneities or susceptibility variations, coupled with a long readout time, are the major sources of this artifact. Correction methods based on a priori off-resonance information such as an acquired field map have been proposed in the literature. An alternative approach estimates the spatially varying off-resonance frequency from the data itself before applying a correction. In this latter approach there is a trade-off between the extent of correction and the chance of increased artifact due to estimation error. This paper introduces an improved algorithm for field map estimation which is both faster and more robust than the existing method. It uses a multi-stage estimation of the field map, starting from a coarse estimate both in frequency and space and proceeds towards higher resolution. The new algorithm is applied to phantom and in vivo images acquired with radial and spiral sequences to give sharper images.

Multifrequency interpolation for fast off-resonance correction

Field inhomogeneities or susceptibility variations produce blurring in images acquired using non-2DFT k-space readout trajectories. This problem is more pronounced for sequences with long readout times such as spiral imaging. Theoretical and practical correction methods based on an acquired field map have been reported in the past. This paper introduces a new correction method based on the existing concept of frequency segmented correction but which is faster and theoretically more accurate. It consists of reconstructing the data at several frequencies to form a set of base images that are then added together with spatially varying linear coefficients derived from the field map. The new algorithm is applied to phantom and in vivo images acquired with projection reconstruction and spiral sequences, yielding sharply focused images.

Air-coupled Lamb wave tomography

An entirely air-coupled inspection system using a pair of micromachined silicon capacitance transducers has been used to image defects in thin plates of different materials (0.7 mm to 2.22 mm thick) using air-coupled Lamb wave tomography. A filtered back projection algorithm was used in a form of difference tomography to reconstruct images of defects up to 10 mm diameter machined in aluminium and perspex (Plexiglas) plates, as well as in samples of carbon fiber reinforced polymer (CFRP). The technique was able to resolve non-central defects as well as multiple flaws within the scan area. This flexible tomographic system was able to produce images of the change in a variety of different acoustic variables from only one set of experimental data, with success dependent on the size, shape, and location of the defect in the scan area.

VOIR: a volumetric image reconstruction algorithm based on Fourier techniques for inversion of the 3-D Radon transform

A novel volumetric image reconstruction algorithm known as VOIR is presented for inversion of the 3-D Radon transform or its radial derivative. The algorithm is a direct implementation of the projection slice theorem for plane integrals. It generalizes one of the most successful methods in 2-D Fourier image reconstruction involving concentric-square rasters to 3-D; in VOIR, the spectral data, which is calculated by fast Fourier techniques, lie on concentric cubes and are interpolated by a bilinear method on the sides of these concentric cubes. The algorithm has great computational advantages over filtered-backprojection algorithms; for images of side dimension N, the numerical complexity of VOIR is O(N(3) log N) instead of O(N (4)) for backprojection techniques. An evaluation of the image processing performance is reported by comparison of reconstructed images from simulated cone-beam scans of a contrast and resolution test object. The image processing performance is also characterized by an analysis of the edge response from the reconstructed images.

Optimization of beam orientations in radiation therapy: some theoretical considerations

The problem of optimizing beam orientations for irradiations with multiple fixed beams is investigated. It is shown that this is a complex, in mathematical terms 'non-convex', optimization problem, whose solution requires sophisticated techniques. In this work, the optimization is performed with the method of simulated annealing. In order to keep the calculation time within reasonable limits, the problem is expressed in the spatial frequency domain using Parseval's theorem. All calculations are then performed in the frequency domain. The algorithm is described in detail. Various treatment techniques, including intensity modulation, are considered. The results for various exemplary cases are presented. They are based on a simplified dose calculation model. A general conclusion is that the optimum beam configuration for multiple-beam irradiations (with more than three beams) tends to be an even distribution over an angular range of 0 to 2 pi.

Array imaging with beam-steered data

The author presents a system model and inversion for the beam-steered data obtained by linearly varying the relative phase among the elements of an array, also known as phased array scan data. The system model and inversion incorporate the radiation pattern of the array's elements. The inversion method utilizes the time samples of the echoed signals for each scan angle instead of range focusing. It is shown that the temporal Fourier transform of the phased array scan data provides the distribution of the spatial Fourier transform of the reflectivity function for the medium to be imaged. The extent of this coverage is related to the array's length and the temporal frequency bandwidth of the transmitted pulsed signal. Sampling constraints and reconstruction procedure for the imaging system are discussed. It is shown that the imaging information obtained by the inversion of phased array scan data is equivalent to the image reconstructed from its synthesized array counterpart.

Image recovery in computer tomography from partial fan-beam data by convex projections

For the image recovery process the authors use the convex projections method, also known as the method of projections onto convex sets (POCS). Several incomplete-data geometries, including those associated with limited source travel and beam-blocking internal opacities, are considered. To enable the recovery several prior-knowledge constraints including one associated with the directivity of the image vector are used. The overall recovery algorithm can be practically implemented by exploiting the Toeplitz structure of key operators.

Selection of a convolution function for Fourier inversion using gridding [computerised tomography application]

In the technique known as gridding, the data samples are weighted for sampling density and convolved with a finite kernel, then resampled on a grid preparatory to a fast Fourier transform. The authors compare the artifact introduced into the image for various convolving functions of different sizes, including the Kaiser-Bessel window and the zero-order prolate spheroidal wave function (PSWF). They also show a convolving function that improves upon the PSWF in some circumstances.

One-step image reconstruction from incomplete data in computer tomography

The authors develop a projection operator that simultaneously projects onto the set of all functions satisfying raysum constraints in parallel-beam CT. The projector can be directly extended to the fan-beam case through the process of rebinning. The projector generates a solution that is closest to the initial estimate among all the functions that are consistent with the available raysum data. It realizes the ART (algebraic reconstruction technique) in one step. The projector furnishes the one-step projection reconstruction (OSPR) for any arbitrary configurations of missing data. Because the projection is one-step, there can be a significant reduction in the number of online computations and memory requirements, especially when the missing data exhibit some pattern within a view or between views.

A Hankel transform approach to tomographic image reconstruction

A relatively unexplored algorithm is developed for reconstructing a two-dimensional image from a finite set of its sampled projections. The algorithm, referred to as the Hankel-transform-reconstruction (HTR) algorithm, is polar-coordinate based. The algorithm expands the polar-form Fourier transform F(r,theta) of an image into a Fourier series in theta calculates the appropriately ordered Hankel transform of the coefficients of this series, giving the coefficients for the Fourier series of the polar-form image f(p,phi); resolves this series, giving a polar-form reconstruction; and interpolates this reconstruction to a rectilinear grid. The HTR algorithm is outlined, and it is shown that its performance compares favorably to the popular convolution-backprojection algorithm.

Direct fourier reconstruction in fan-beam tomography

We consider the problem of reconstructing tomographic imagery from fan-beam projections using the direct Fourier method (DFM). Previous DFM reconstructions from parallel-beam projections produced images of quality comparable to that of filtered convolution back-projection. Moreover, the number of operations using DFM in the parallel-beam case is proportional to N2 log N versus N3 for back projection [3]. The fan-beam case is more complicated because additional interpolation of the nonuniformly spaced rebinned data is required. We derive bounds on the detector spacing in fan-beam CT that enable direct Fourier reconstruction and describe the full algorithm necessary for processing the fan-beam data. The feasibility of the method is demonstrated with an example. A key result of this paper is that high-quality imagery can be reconstructed from fan-beam data using the DFM in 0 (N2 log N) operations.

Errors in reprojection methods in computenzed tomography

Iterative tomographic reconstruction methods have been developed which can enforce various physical constraints on the reconstructed image. An integral part of most of these methods is the repro. jection of the reconstructed image. These estimated projections are compared to the original projection data and modified according to some criteria based on a priori constraints. In this paper, the errors generated by such reprojection schemes are investigated. Bounds for these errors are derived under simple signal energy assumptions and using probabilistic assumptions on the distribution of discontinuities. These bounds can be used in the enforcement of constraints, in the determination of convergence of the iterative methods, and in the detection of artifacts.

Interpolation from samples on a linear spiral scan

An interpolation method useful for reconstructing an image from its Fourier plane samples on a linear spiral scan trajectory is presented. This kind of sampling arises in NMR imaging. We first present a theorem that enables exact interpolation from spiral samples to a Cartesian lattice. We then investigate two practical implementations of the theorem in which a finite number of interpolating points are used to calculate the value at a new point. Our experimental results confirm the theorem's validity and also demonstrate that both practical implementations yield very good reconstructions. Thus, the theorem and/or its practical implementations suggest the possibility of using direct Fourier reconstruction from linear spiral-scan NMR imaging.

A fast sinc function gridding algorithm for fourier inversion in computer tomography

The Fourier inversion method for reconstruction of images in computerized tomography has not been widely used owing to the perceived difficulty of interpolating from polar or other measurement grids to the Cartesian grid required for fast numerical Fourier inversion. Although the Fourier inversion method is recognized as being computationally faster than the back-projection method for parallel ray projection data, the artifacts resulting from inaccurate interpolation have generally limited application of the method. This paper presents a computationally efficient gridding algorithm which can be used with direct Fourier transformation to achieve arbitrarily small artifact levels. The method has potential for application to other measurement geometries such as fan-beam projections and diffraction tomography and NMR imaging.